Inequalities

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These are the solutions to the NSSCO past questions on Inequalities.
The TI-84 Plus CE shall be used for applicable questions.
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Notes and Rules

(1.) At least 5 means \ge 5
It means that the minimum should be 5

(2.) At most 5 means \le 5
It means that the maximum should be 5

(3.) Just 5 means = 5
It means exactly 5 or equal to 5

(4.) More than 5 means \gt 5

(5.) Less than 5 means \lt 5

(6.) No more than 5 means \le 5
It should not be more than 5
It means 5 or less

(7.) No less than 5 means \ge 5
It should not be less than 5
It means 5 or more

Rules of Inequalities
c, d, e are real numbers

$ (1.)\:\:If\:\: c \lt d \:\:and\:\: d \lt e, \:\:then\:\: c \lt e ...Transitive\:\:Rule \\[3ex] (2.)\:\:If\:\: c \gt d \:\:and\:\: d \gt e, \:\:then\:\: c \gt e ...Transitive\:\:Rule \\[3ex] (3.)\:\:If\:\: c \lt d, \:\:then\:\: d \gt c \\[3ex] (4.)\:\:If\:\: c \gt d, \:\:then\:\: d \lt c \\[3ex] (5.)\:\:If\:\: c \lt d, \:\:then\:\: -c \gt -d \\[3ex] (6.)\:\:If\:\: c \gt d, \:\:then\:\: -c \lt -d \\[3ex] (7.)\:\:If\:\: c \lt d, \:\:then\:\: \dfrac{1}{c} \gt \dfrac{1}{d} \\[5ex] (8.)\:\:If\:\: c \gt d, \:\:then\:\: \dfrac{1}{c} \lt \dfrac{1}{d} \\[5ex] (9.)\:\:If\:\: c \lt d, \:\:then\:\: (c + e) \lt (d + e) \\[3ex] (10.)\:\:If\:\: c \gt d, \:\:then\:\: (c + e) \gt (d + e) \\[3ex] (11.)\:\:If\:\: c \lt d, \:\:then\:\: (c - e) \lt (d - e) \\[3ex] (12.)\:\:If\:\: c \gt d, \:\:then\:\: (c - e) \gt (d - e) \\[3ex] (13.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \lt de \\[3ex] (14.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \gt de \\[3ex] (15.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \gt de \\[3ex] (16.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \lt de \\[3ex] (17.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} \\[5ex] (18.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (19.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (20.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} $

(2.) Solve the inequality $8(2 - 3x) - 4(1 - 2x) \le 0$

$ 8(2 - 3x) - 4(1 - 2x) \le 0 \\[3ex] 16 - 24x - 4 + 8x \le 0 \\[3ex] -16x + 12 \le 0 \\[3ex] -16x \le 0 - 12 \\[3ex] -16x \le -12 \\[3ex] x \ge \dfrac{-12}{-16} \\[5ex] ...\text{inequality is reversed...division by a negative value} \\[5ex] x \ge \dfrac{3}{4} $

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